• Previous Article
    Finite-time synchronization of competitive neural networks with mixed delays
  • DCDS-B Home
  • This Issue
  • Next Article
    Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux
December  2016, 21(10): 3637-3654. doi: 10.3934/dcdsb.2016114

Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays

1. 

Vietnam Education Publishing House, 81 Tran Hung Dao, Hanoi, Vietnam

2. 

Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

3. 

Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam

Received  November 2015 Revised  August 2016 Published  November 2016

We deal with the Cauchy problem associated with integro-differential inclusions of diffusion-wave type involving infinite delays. Based on the behavior of resolvent operator associated with the linear part, an explicit estimate for solutions will be established. As a consequence, the weak stability of zero solution is proved in case the resolvent operator is asymptotically stable.
Citation: Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3637-3654. doi: 10.3934/dcdsb.2016114
References:
[1]

B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semi-linear Cauchy problems with non-dense domain,, Nonlinear Anal., 72 (2010), 3190.  doi: 10.1016/j.na.2009.12.016.  Google Scholar

[2]

N. T. Anh and T. D. Ke, Decay integral solutions for neutral fractional differential equations with infinite delays,, Math. Methods Appl. Sci., 38 (2015), 1601.  doi: 10.1002/mma.3172.  Google Scholar

[3]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces,, Ph.D. Thesis, (2001).   Google Scholar

[4]

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations,, Dover Publications, (2006).   Google Scholar

[5]

T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations,, Dyn. Syst. Appl., 10 (2001), 89.   Google Scholar

[6]

J. P. Carvalho dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations,, Appl. Math. Lett., 23 (2010), 960.  doi: 10.1016/j.aml.2010.04.016.  Google Scholar

[7]

N. M. Chuong, T. D. Ke and N. N. Quan, Stability for a class of fractional partial integro-differential equations,, J. Integral Equations Appl., 26 (2014), 145.  doi: 10.1216/JIE-2014-26-2-145.  Google Scholar

[8]

C. Cuevas and J. César de Souza, Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay,, Nonlinear Anal., 72 (2010), 1683.  doi: 10.1016/j.na.2009.09.007.  Google Scholar

[9]

C. Cuevas and J. César de Souza, $S$-asymptotically $\omega$-periodic solutions of semilinear fractional integro-differential equations,, Appl. Math. Lett., 22 (2009), 865.  doi: 10.1016/j.aml.2008.07.013.  Google Scholar

[10]

E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations,, Discrete Contin. Dyn. Syst., (2007), 277.   Google Scholar

[11]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$,, Proc. Amer. Math. Soc., 118 (1993), 447.  doi: 10.2307/2160321.  Google Scholar

[12]

R. D. Driver, Ordinary and Delay Differential Equations,, Springer-Verlag, (1977).   Google Scholar

[13]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Society for Industrial and Applied Mathematics (SIAM), (1999).  doi: 10.1137/1.9781611971088.  Google Scholar

[14]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Translated from the Russian. Mathematics and its Applications (Soviet Series), (1988).  doi: 10.1007/978-94-015-7793-9.  Google Scholar

[15]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation,, Osaka J. Math., 27 (1990), 309.   Google Scholar

[16]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II,, Osaka J. Math., 27 (1990), 797.   Google Scholar

[17]

C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay,, Nonlinear Anal., 51 (2002), 765.  doi: 10.1016/S0362-546X(01)00861-6.  Google Scholar

[18]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar

[19]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[20]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, Lecture Notes in Mathematics, (1473).  doi: 10.1007/BFb0084432.  Google Scholar

[21]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces,, in: de Gruyter Series in Nonlinear Analysis and Applications, (2001).  doi: 10.1515/9783110870893.  Google Scholar

[22]

F. Mainardi and P. Paradisi, Fractional diffusive waves,, J. Comput. Acoustics, 9 (2001), 1417.  doi: 10.1142/S0218396X01000826.  Google Scholar

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Phys. Rep., 339 (2000), 1.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[24]

R. Metzler and J. Klafter, Accelerating Brownian motion: A fractional dynamics approach to fast diffusion,, Europhys. Lett., 51 (2000), 492.  doi: 10.1209/epl/i2000-00364-5.  Google Scholar

[25]

J.-S. Pang and D. E. Stewart, Differential variational inequalities,, Math. Program. Ser. A, 113 (2008), 345.  doi: 10.1007/s10107-006-0052-x.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems,, J. Math. Anal. Appl., 382 (2011), 426.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[28]

T. I. Seidman, Invariance of the reachable set under nonlinear perturbations,, SIAM J. Control Optim., 25 (1987), 1173.  doi: 10.1137/0325064.  Google Scholar

show all references

References:
[1]

B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semi-linear Cauchy problems with non-dense domain,, Nonlinear Anal., 72 (2010), 3190.  doi: 10.1016/j.na.2009.12.016.  Google Scholar

[2]

N. T. Anh and T. D. Ke, Decay integral solutions for neutral fractional differential equations with infinite delays,, Math. Methods Appl. Sci., 38 (2015), 1601.  doi: 10.1002/mma.3172.  Google Scholar

[3]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces,, Ph.D. Thesis, (2001).   Google Scholar

[4]

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations,, Dover Publications, (2006).   Google Scholar

[5]

T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations,, Dyn. Syst. Appl., 10 (2001), 89.   Google Scholar

[6]

J. P. Carvalho dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations,, Appl. Math. Lett., 23 (2010), 960.  doi: 10.1016/j.aml.2010.04.016.  Google Scholar

[7]

N. M. Chuong, T. D. Ke and N. N. Quan, Stability for a class of fractional partial integro-differential equations,, J. Integral Equations Appl., 26 (2014), 145.  doi: 10.1216/JIE-2014-26-2-145.  Google Scholar

[8]

C. Cuevas and J. César de Souza, Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay,, Nonlinear Anal., 72 (2010), 1683.  doi: 10.1016/j.na.2009.09.007.  Google Scholar

[9]

C. Cuevas and J. César de Souza, $S$-asymptotically $\omega$-periodic solutions of semilinear fractional integro-differential equations,, Appl. Math. Lett., 22 (2009), 865.  doi: 10.1016/j.aml.2008.07.013.  Google Scholar

[10]

E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations,, Discrete Contin. Dyn. Syst., (2007), 277.   Google Scholar

[11]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$,, Proc. Amer. Math. Soc., 118 (1993), 447.  doi: 10.2307/2160321.  Google Scholar

[12]

R. D. Driver, Ordinary and Delay Differential Equations,, Springer-Verlag, (1977).   Google Scholar

[13]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Society for Industrial and Applied Mathematics (SIAM), (1999).  doi: 10.1137/1.9781611971088.  Google Scholar

[14]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Translated from the Russian. Mathematics and its Applications (Soviet Series), (1988).  doi: 10.1007/978-94-015-7793-9.  Google Scholar

[15]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation,, Osaka J. Math., 27 (1990), 309.   Google Scholar

[16]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II,, Osaka J. Math., 27 (1990), 797.   Google Scholar

[17]

C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay,, Nonlinear Anal., 51 (2002), 765.  doi: 10.1016/S0362-546X(01)00861-6.  Google Scholar

[18]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar

[19]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[20]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, Lecture Notes in Mathematics, (1473).  doi: 10.1007/BFb0084432.  Google Scholar

[21]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces,, in: de Gruyter Series in Nonlinear Analysis and Applications, (2001).  doi: 10.1515/9783110870893.  Google Scholar

[22]

F. Mainardi and P. Paradisi, Fractional diffusive waves,, J. Comput. Acoustics, 9 (2001), 1417.  doi: 10.1142/S0218396X01000826.  Google Scholar

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Phys. Rep., 339 (2000), 1.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[24]

R. Metzler and J. Klafter, Accelerating Brownian motion: A fractional dynamics approach to fast diffusion,, Europhys. Lett., 51 (2000), 492.  doi: 10.1209/epl/i2000-00364-5.  Google Scholar

[25]

J.-S. Pang and D. E. Stewart, Differential variational inequalities,, Math. Program. Ser. A, 113 (2008), 345.  doi: 10.1007/s10107-006-0052-x.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems,, J. Math. Anal. Appl., 382 (2011), 426.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[28]

T. I. Seidman, Invariance of the reachable set under nonlinear perturbations,, SIAM J. Control Optim., 25 (1987), 1173.  doi: 10.1137/0325064.  Google Scholar

[1]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[2]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[3]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[4]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[5]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[6]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[7]

Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011

[8]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[9]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[10]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[11]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[12]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[13]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[14]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[15]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[16]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[17]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[18]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[19]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[20]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]